diff --git a/array_api_compat/common/_aliases.py b/array_api_compat/common/_aliases.py
index 167c7c0f..8792aa2e 100644
--- a/array_api_compat/common/_aliases.py
+++ b/array_api_compat/common/_aliases.py
@@ -489,10 +489,7 @@ def tensordot(x1: ndarray,
     return xp.tensordot(x1, x2, axes=axes, **kwargs)
 
 def vecdot(x1: ndarray, x2: ndarray, /, xp, *, axis: int = -1) -> ndarray:
-    ndim = max(x1.ndim, x2.ndim)
-    x1_shape = (1,)*(ndim - x1.ndim) + tuple(x1.shape)
-    x2_shape = (1,)*(ndim - x2.ndim) + tuple(x2.shape)
-    if x1_shape[axis] != x2_shape[axis]:
+    if x1.shape[axis] != x2.shape[axis]:
         raise ValueError("x1 and x2 must have the same size along the given axis")
 
     if hasattr(xp, 'broadcast_tensors'):
@@ -500,9 +497,9 @@ def vecdot(x1: ndarray, x2: ndarray, /, xp, *, axis: int = -1) -> ndarray:
     else:
         _broadcast = xp.broadcast_arrays
 
-    x1_, x2_ = _broadcast(x1, x2)
-    x1_ = xp.moveaxis(x1_, axis, -1)
-    x2_ = xp.moveaxis(x2_, axis, -1)
+    x1_ = xp.moveaxis(x1, axis, -1)
+    x2_ = xp.moveaxis(x2, axis, -1)
+    x1_, x2_ = _broadcast(x1_, x2_)
 
     res = x1_[..., None, :] @ x2_[..., None]
     return res[..., 0, 0]
diff --git a/array_api_compat/torch/linalg.py b/array_api_compat/torch/linalg.py
index 63f0135b..7e7e2415 100644
--- a/array_api_compat/torch/linalg.py
+++ b/array_api_compat/torch/linalg.py
@@ -5,7 +5,8 @@
     import torch
     array = torch.Tensor
     from torch import dtype as Dtype
-    from typing import Optional
+    from typing import Optional, Union, Tuple, Literal
+    inf = float('inf')
 
 from ._aliases import _fix_promotion, sum
 
@@ -23,8 +24,16 @@
 
 # Note: torch.linalg.cross does not default to axis=-1 (it defaults to the
 # first axis with size 3), see https://github.com/pytorch/pytorch/issues/58743
+
+# torch.cross also does not support broadcasting when it would add new
+# dimensions https://github.com/pytorch/pytorch/issues/39656
 def cross(x1: array, x2: array, /, *, axis: int = -1) -> array:
     x1, x2 = _fix_promotion(x1, x2, only_scalar=False)
+    if not (-min(x1.ndim, x2.ndim) <= axis < max(x1.ndim, x2.ndim)):
+        raise ValueError(f"axis {axis} out of bounds for cross product of arrays with shapes {x1.shape} and {x2.shape}")
+    if not (x1.shape[axis] == x2.shape[axis] == 3):
+        raise ValueError(f"cross product axis must have size 3, got {x1.shape[axis]} and {x2.shape[axis]}")
+    x1, x2 = torch.broadcast_tensors(x1, x2)
     return torch_linalg.cross(x1, x2, dim=axis)
 
 def vecdot(x1: array, x2: array, /, *, axis: int = -1, **kwargs) -> array:
@@ -32,19 +41,18 @@ def vecdot(x1: array, x2: array, /, *, axis: int = -1, **kwargs) -> array:
 
     x1, x2 = _fix_promotion(x1, x2, only_scalar=False)
 
+    # torch.linalg.vecdot incorrectly allows broadcasting along the contracted dimension
+    if x1.shape[axis] != x2.shape[axis]:
+        raise ValueError("x1 and x2 must have the same size along the given axis")
+
     # torch.linalg.vecdot doesn't support integer dtypes
     if isdtype(x1.dtype, 'integral') or isdtype(x2.dtype, 'integral'):
         if kwargs:
             raise RuntimeError("vecdot kwargs not supported for integral dtypes")
-        ndim = max(x1.ndim, x2.ndim)
-        x1_shape = (1,)*(ndim - x1.ndim) + tuple(x1.shape)
-        x2_shape = (1,)*(ndim - x2.ndim) + tuple(x2.shape)
-        if x1_shape[axis] != x2_shape[axis]:
-            raise ValueError("x1 and x2 must have the same size along the given axis")
 
-        x1_, x2_ = torch.broadcast_tensors(x1, x2)
-        x1_ = torch.moveaxis(x1_, axis, -1)
-        x2_ = torch.moveaxis(x2_, axis, -1)
+        x1_ = torch.moveaxis(x1, axis, -1)
+        x2_ = torch.moveaxis(x2, axis, -1)
+        x1_, x2_ = torch.broadcast_tensors(x1_, x2_)
 
         res = x1_[..., None, :] @ x2_[..., None]
         return res[..., 0, 0]
@@ -59,8 +67,22 @@ def trace(x: array, /, *, offset: int = 0, dtype: Optional[Dtype] = None) -> arr
     # Use our wrapped sum to make sure it does upcasting correctly
     return sum(torch.diagonal(x, offset=offset, dim1=-2, dim2=-1), axis=-1, dtype=dtype)
 
-__all__ = linalg_all + ['outer', 'trace', 'matmul', 'matrix_transpose', 'tensordot',
-                        'vecdot', 'solve']
+def vector_norm(
+    x: array,
+    /,
+    *,
+    axis: Optional[Union[int, Tuple[int, ...]]] = None,
+    keepdims: bool = False,
+    ord: Union[int, float, Literal[inf, -inf]] = 2,
+    **kwargs,
+) -> array:
+    # torch.vector_norm incorrectly treats axis=() the same as axis=None
+    if axis == ():
+        keepdims = True
+    return torch.linalg.vector_norm(x, ord=ord, axis=axis, keepdim=keepdims, **kwargs)
+
+__all__ = linalg_all + ['outer', 'matmul', 'matrix_transpose', 'tensordot',
+                        'cross', 'vecdot', 'solve', 'trace', 'vector_norm']
 
 _all_ignore = ['torch_linalg', 'sum']